J17

Exercise
Factorize:
1. x 2 + 7x + 10 2. x 2 + 13x + 42
3. x 2 + 9x + 14 4. x 2 + 2x – 15
5. x 2 + x – 12 6. p 2 – 10p + 12
7. a 2 – 2a – 35 8. d 2 + 5d – 36
9. v 2 – 14v + 45 10. u 2 – u – 56
11. x 2 + 9x + 8 12. n 2 – 10n + 25
13. x 2 – 2x + 1 14. c 2 – c – 2
15. y 2 + 14y + 24 16. w 2 + 5w – 6
17. 9 – 6r + r 2 18. 49 + 14t + t 2
19. 4 + 4k + k 2 20. x 2 + 11x – 26
The difference of two squares
Earlier in this chapter, we learnt that (a + b)(a – b) = a 2 – b 2 . Thus, the difference of the
squares of two numbers is equal to the product of their sum and their difference. The
factors of a 2 – b 2 are (a + b) and (a – b).
In order to factorize the difference of two squares, it is important to rewrite the
expression so that the squares are clearly seen. For example:
(a) a 2 – 9b 2 = a 2 – (3b) 2
= (a – 3b)(a + 3b),
(b) 1 – 16x 2 = 1 2 – (4x) 2
= (1 + 4x)(1 – 4x),
(c) 4x 2 – 25y 2 = (2x) 2 – (5y) 2
= (2x + 5y)(2x – 5y).
However, some expressions which involve the difference of squares require us to find
the common factors first.
Example
Factorize:
(a) 7x 2 – 7y 2 (b) 3x 2 – 75y 2
Solutions
(a)
(b)
Both terms have a common factor, 7. Therefore factorize as follows:
7x 2 – 7y 2 = 7(x 2 – y 2 )
= 7(x + y(x – y)
3 is a common factor in the two terms.
Thus, 3x 2 – 75y 2 = 3(x 2 – 25y 2 )
= 3[x 2 – (5y) 2 ]